Path axis

Kepler railways and rail axes

Orbital axes are the symmetry axes of closed elliptical orbits ( Kepler orbits ) in astronomy . The major semi-axis is the distance of the main vertex from the center of the ellipse and thus the greatest distance that can be laid in an ellipse from the center. ${\ displaystyle a}$

Significance for the path calculation

According to Kepler's first law , a celestial body does not run around the center of the ellipse (halfway between the vertices), but around one of the two focal points . Therefore, its orbit is generally described in a coordinate system whose origin lies in the focal point with the center of mass ( orbit coordinate system ). The distance to the origin is then described by the radius vector .

How much the distance between the rotating body and the origin of the coordinates changes on the orbit depends primarily on the eccentricity of the orbit ellipse:

the minimum distance is calculated from large semi-axis minus linear eccentricity , , ${\ displaystyle a-r _ {\ mathrm {min}}}$
the maximum value . ${\ displaystyle a + r _ {\ mathrm {min}}}$

With decreasing eccentricity with unchanged semi-axis, the center of the ellipse, in the drawing at (2 | 0), and the second focal point, in the drawing at (4 | 0), move closer and closer to the first focal point (0 | 0). In the borderline case , the path is a circle that has the same axis or radius as the ellipse shown (namely 3 units of length). On these two related orbits, a small body orbits a large mass at the focal point at (0 | 0) according to Kepler's third law with the same orbital period . ${\ displaystyle r _ {\ mathrm {min}} = 0}$

Influence of the barycentre

The Kepler planetary theory is an idealization that neglects the gravitation of the smaller on the larger body. In reality, both orbit a common center of gravity, called the barycenter of the system in celestial mechanics .

Example of the lunar orbit (average):
closest point to earth ( perigee )	362,102 km
most distant point ( apogee )	404,694 km
Mean value ( major semi-axis )	383,398 km
Major semiaxis of the lunar orbit	378,739 km
Major semiaxis of the earth's orbit	004,659 km

The last two values relate to the movement of the earth and moon around the earth-moon center of gravity (EMS). Since the moon has about 1/81 the mass of the earth , the EMS is on average 4700 km away from the center of the earth (earth's orbit axis around the EMS), i.e. about 1700 km below the earth's surface . If the moon is far from the earth, the EMS is further away from the center of the earth; if the moon is close to the earth, the distance between the center of the earth and the EMS is also smaller. This fluctuation remains below a few hundred kilometers.

In the case of the moons of other planets, this difference is hardly apparent, as their relative masses are much lower. Here you can take the mean value of the two extreme values as the orbit axis, which Kepler already referred to as “mean distance”.

The barycentric difference is also minimal in the planetary orbits - with the exception of the "giant planets" Jupiter and Saturn , which have about 1.0 ‰ and 0.3 ‰ of the solar mass .

Remarks

↑ In mathematics denotes the numerical eccentricity, in astronomy it is specified as, it lies in the interval . The linear eccentricity, mathematically , a measure of length, e.g. B. in kilometers or astronomical units , denotes the distance between the focal point and the center of the ellipse. ${\ displaystyle \ epsilon}$ ${\ displaystyle e}$ ${\ displaystyle [0.1 [}$ ${\ displaystyle e}$ ${\ displaystyle r _ {\ mathrm {min}}}$

[Anm.-1] In mathematics denotes the numerical eccentricity, in astronomy it is specified as, it lies in the interval . The linear eccentricity, mathematically , a measure of length, e.g. B. in kilometers or astronomical units , denotes the distance between the focal point and the center of the ellipse. ${\ displaystyle \ epsilon}$ ${\ displaystyle e}$ ${\ displaystyle [0.1 [}$ ${\ displaystyle e}$ ${\ displaystyle r _ {\ mathrm {min}}}$